All of the 3rd grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$6.50$ each for teachers and $$3.00$ each for students, and the group paid $$31.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$26.00$ each for teachers and $$8.50$ each for students, and the group paid $$103.00$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+3y = 31}$ ${26x+8.5y = 103}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-26x-12y = -124}$ ${26x+8.5y = 103}$ Add the top and bottom equations together. $ -3.5y = -21 $ $ y = \dfrac{-21}{-3.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {6.5x+3y = 31}$ to find $x$ ${6.5x + 3}{(6)}{= 31}$ $6.5x+18 = 31$ $6.5x = 13$ $x = \dfrac{13}{6.5}$ ${x = 2}$ You can also plug ${y = 6}$ into $ {26x+8.5y = 103}$ and get the same answer for $x$ ${26x + 8.5}{(6)}{= 103}$ ${x = 2}$ There were $2$ teachers and $6$ students on the field trips.